Chapter 2 Projections and Unitary elements

The K-theory of a -algebra is defined in terms of equivalence classes of its projections and equivalence classes of its unitary elements — possibly after adjoining a unit and forming matrix algebras.
In this chapter we derive the facts needed about projections and unitary elements with emphasis on the equivalence relation defined by homotopy and also — for projections — Murray-von Neumann equivalence and unitary equivalence.

2.1 Homotopy classes of unitary elements

2.1.1 Homotopy

Let be a topological space say that two points are homotopic in written in if there exists a continuous function such that and . The relation is an equivalence relation on . We call a continuous path from to .

Definition 2.1.2.
Let be a unital C-star-Algebra. Recall that an element is unitary if . Denote the group of unitary elements in by . From above we know that is an equivalence relation on . Now define the set of all such that to be .

Products of homotopic unitaries

If are unitary elements in a -algebra with
and , then .
Indeed, find continuous paths in from to for Then is a continuous path in from to .

Recall in (2) below that the spectrum of a unitary element is contained in . The unit circle.

Lemma 2.1.3
Let be a unital C-star-Algebra, then

For each self-adjoint element ,
If is a unitary element in with , then .
If and are unitary elements in with then

Proof of 1:
It follows from the 2.1.13 Theorem (Continuous functional calculus) That if is continuous and if is self-adjoint in , then This is only possible because on inverse and conjugate coincide.
This implies that is a unitary, and in particular is a unitary element in .
For each define via .

Since is continuous, so is the path in .
Since the map is continuous for t at every x, in particular it should be continuous at just h
Therefore .

Proof of 2:
If , then does not belong to for some real number . Let be the real function on the spectrum of defined by , where .
Then is continuous, and for every .
Set . It follows that is a self-adjoint element in with , and so belongs to by part 1.
The main idea here is that if the element has as spectrum only part of the unit circle, we can define a branch on the argument function acting on the spectrum of the element, from there we can access our element as an argument of some self adjoint element on the unit circle, then we apply part 1 to this exponential.

Proof of 3:
If , then because every unitary is of norm 1. This implies that . (Spectral radius and norm are equal for self-adjoints)
This in turn implies that . Now we summon part 2 to conclude that and finally that as desired.
This proof exploits the relationship between the spectral radius of a normal element and its norm, along with the convenient fact that
and .

Each unitary in has finite spectra and therefore belongs to by Lemma 2.1.3 part 2. This proves the following corollary:

Corollary 2.1.4
The unitary group in is connected, in other words

Lemma 2.1.5 (Whitehead)
Let be a unital C-star-Algebra and let . Then

In .

It follows in particular that
in

Proof:

From Lemma 2.1.3 we get that
Hence The other claims follow in a similar way.

Proposition 2.1.6.
Let be a unital C-star-Algebra

is normal subgroup of
is open and closed relative to
An element belongs to if and only if For some natural number and some self-adjoint elements
Note first that is closed under multiplication. (Why? See the comment underneath Definition 2.1.2.)
We must also show that if belongs to then so do and for every . (Normality)
Let be a continuous path in from to .
Then and are continuous paths in from to and to respectively. Not exactly sure how continuity is immediate, perhaps since each point in the path is a unitary, we can continuously take inverses and also conjugate by unitaries
Taking products is continuous and inverse map is differentiable, think about the tools needed to prove non empty spectrum theorem.
This proves 1.

Let be the set of elements in of form Where and are all self-adjoint elements in .
It follows from 1. and Lemma 2.1.3 (1), that . Easily
Since for every self-adjoint , we see that is a Group.

Take in and with . Then and by 2.1.3 Lemma (3) and its proof for some self-adjoint element . Hence This shows that is open relative to .
The complement is a disjoint union of cosets of the form with . Each of these cosets are homeomorphic to and therefore open relative to , consequently is closed in .
In conclusion, is a non-empty subse of , is closed and open in , and is connected. This implies that .
Therefore 2. and 3. hold.

Lemma 2.1.7.
Let and be unital -algebras, and let be a surjective (And hence unit preserving) star-homomorphism.

For each , there exists such that
Where is the map induced by .
If is a unitary element in , and if there is a unitary element such that , then belongs to
in other words lifts to a unitary element in .

Proof of 1:
A unital -homomorphism is continuous and maps unitaries to unitaries. Therefore .
Conversely, if then by Proposition 2.1.6. for some self-adjoint elements in . Since is surjective, there are elements such that Put
Then and
set
Then belongs to by Proposition 2.1.6. and .
Reflection:
The first containment follows from continuity of the map.
The second relies on surjectivity, we have this decomposition of elements in into products of exponentials of self-adjoint elements, we can find elements who map onto them, we then. take the real part of each of the ’s, this gives us self-adjoint elements we need in order to represent the pre image as living in (A).

Proof 2:
Follows immediately from 1. and Lemma 2.1.5 (Whitehead).
[Insert detailed rumination]

Proof 3:
If , then belongs to , and hence for some by 1. Now as desired.

Let be a unital C-star-Algebra. The group of invertible elements in is denoted by , and the set of elements in which are homotopic, denote it by

The group of unitary elements in is a subgroup of

Part 2. of the following proposition says that is a retract of
A subspace of a topological space is called a retract of if there is continuous satisfying
in for all , and for all .

For each element , set The element is called the absolute value.

Proposition 2.1.8
Let be a unital C-star-Algebra

If is an invertible element in , then so is , and belongs to . Clearly, .
The normalization of an invertible element is unitary.
The map defined in 1. is continuous for every , and in for every .
Normalizing is continuous over the unitaries, moreover, the normalization of z is homotopic to z in the set of invertible of A.
If are unitary elements in , and in , then in
If two unitaries are homotopic in the invertibles, we can strengthen this condition to homotopic inside the unitaries.
Proof of 1:
If then and by closure
It follows that is invertible. (WHY?_Perhaps the continuity of the square root?This follows from the spectral theorem, so take z, map it to the identity on the spectrum, it must be invertible, so zero cannot be in the spectrum, now take the square root, and it is still invertible)
Put
Then Moreover is invertible, since it is a product of two invertible elements and , since
Proof of 2:

Multiplication in a -algebra is continuous, and so is the map in . To show that is continuous, it therefore suffices to show that the map is continuous. This map is composed of the map and the map for .
The former of these maps is continuous because the involution and multiplication is continuous. It now suffices to show that is continuous on any bounded set. But this follows from Lemma 1.2.5. because each bounded subset of is contained in , where and
—————————-
Aside:
is the set of self-adjoints in with spectrum contained in .
Lemma 1.2.5. Says that any continuous function on , then is continuous.
——————————
If is a unitary element in , then and so .
Let be given, and put for .This is the homotopy between z and its normalization
Then and . Since is positive and invertible, there exists for which .
For each , we have
Thus and are invertible. The map is continuous, and so in

Proof of 3: If is a continuous path in from to , then is a continuous path in from to .

Theorem 2.1.9. (The polar decomposition)
The factorization from Proposition 2.1.8 of an invertible element is called the (unitary) polar decomposition for . We shall often write instead of and the point of emphasis is then that each invertible element in a unital C-star-Algebra written as for a unique unitary

Theorem 2.1.10 (Carl Neumann Series)
Let be a unital C-star-Algebra. Then each with is invertible, and its inverse is given by It follows in particular that the norm of can be estimated byProposition 2.1.11
Let be a unital C-star-Algebra.
Let be an invertible element in , and take with . Then is invertible,and in

2.2 Equivalence of projections

Definition 2.2.1
Recall that an element in a C-star-Algebra is a projection if The set of projections in a -algebra is denoted by . From paragraph 2.1.1 we have the homotopy equivalence relation on . We introduce two additional equivalence relations on :

if there exists such that and
We call p a source projection, and we call q a range projection
(We call this type of equivalence, Murray-von Neumann equivalence, or MvN equiv.)
if there exists a unitary where is the Unitizations of , and we have that .
Remark:
When is a projection, we call it a partial isometry.
If is a projection, then so is .
We call a support projection. We call a range projection.
Put and . Then we have the following much used identities:(see exercise 2.5)

We can use this to show that the Murray-von Neumann relation is transitive.

Proposition 2.2.2
Let be projections in a unital C-star-Algebra . The following conditions are equivalent:

for some unitary
and .
Proof:

Lemma 2.2.3
Let

be a projection in an

algebra

, and let

be a self-adjoint element in

. Put

then

Proof:
Since

is self-adjoint, its spectrum consists of real numbers. Recall that

. It suffices to show that if

is such that

then

.
For such a real number

, the element

is invertible in

and

Not exactly sure how to get this other than via translation
It follows that

It follows that is invertible (This is not hard to see since the spectrum of p is either 0 or 1, so the term on the left ought to be invertible, so just multiply on the left by its inverse and by group closure we have what we need. ), and so this implies that is invertible, which shows that

Proposition 2.2.4
If are projections in a C-star-Algebra and then .
Proof:
Put for Then, is self-adjoint, this is clear.
Secondly, The middle step:and, is continuous. Moreover, in the notation of Lemma 1.2.5
each belongs to (Here Omega_K _denotes the set of self-adjoint elements with spectrum contained in K) by Lemma 2.2.3 where and
Now invoke Urysohn’s Lemma to define to be the continuous function which vanishes on and is equal to on the interval .
Then is a projection for each since .
The path is continuous by Lemma 1.2.5. and hence, In

The next proposition says that if two self-adjoint elements in a unital C* algebra are similar, then they are unitarily equivalent.

Proposition 2.2.5
Let be self-adjoint elements in a unital C-star-Algebra and suppose that for some invertible element .Let be the Theorem 2.1.9. (The polar decomposition) of with then .
Proof:
The equation implies that and because and are self-adjoint, also .
Hence So commutes with . Consequently, commutes with all elements of and so in particular with . It follows that as desired.

Proposition 2.2.6
Let be a -algebra, let Then in if and only if there exists a unitary element homotopic to the identity which implements unitary equivalence,
.

Proof:
Throughout this proof, denotes the unit of .
Suppose first that for some . Let be a continuous map of unitaries in from to .
Then, since is an ideal of is a continuous path of projections in from to .(what?)
(It is clear that for each t, we have a projection, and the path is the desired one, continuity in P(A) requires A to be an ideal I.E, conjugation will stay in A)
Conversely, if , then there exists projections in such that for .(How on earth is it the case that these exist?The homotopy gives us the path,so chop it up.)
It therefore suffices to prove the implication in the case where .
Put then , and Hence is invertible and in by Proposition 2.1.11.
Let be the polar decomposition for , see paragraph 2.19.
Then . From the properties of the polar decomposition we have that inside of , and this entails that belongs to . By Proposition 2.1.8.

Note:
We have three equivalence relations on the set of projections of a C*-algebra. It will be shown in example 2.2.9 that these equivalence relations are different from each other.
That
Homotopy Unitary equiv Murray-von Neumann equiv
Proposition 2.2.8 says that the three relations are actually the same when we pass to matrix algebras.

Proposition 2.2.7
Let be projections in a -algebra .

If , then
If then .
Proof:
Part 1. is an immediate consequence of Proposition 2.2.6.
If for some unitary , then ,
and this proves (ii).

Proposition 2.2.8
Let be projections in a C-star-Algebra .

If , then inside of .
If , then inside of .

2.2.10 Liftings
Suppose that and are -algebras and that is a surjective homomorphism. Given an element , an element is called a lift of if . The set of all lifts of is then the coset . We shall here be concerned with the possibility of lifting an element in with a certain property to an element in with the same property. Along this line we have the following results.

Every element has a lift to an element with .
Every self-adjoint element lifts to a self-adjoint element in . Moreover the self-adjoint lift can be chosen to have the same norm as .
Every positive element lifts to a positive element , moreover, the positive element can be chosen so that it has the same norm as .
A normal element in does not in general lift to a normal element in .
A projection in does not in general lift to a projection in .
Unitaries also do not lift in general to unitaries.

Proof of 2 (lifting self-adjoints):
Let be any lift of , set then is self-adjoint and . (why? lets see the full calculation)

to arrange that the lift has the same norm as , let be the continuous function given by Put Then is a normal element, being a continuous function of a normal element (Not exactly clear how we get a normal element, maybe functional calculus, but for this I thought we needed a self-adjoint element, well above we see that the definition of a_0, forces it to be self-adjoint, which in this case means we get to apply the functional calculus, the normality comes from where then? It is easy to check that the functional calculus preserves normality), and This shows that is self-adjoint, and that because the norm of is the spectral radius. Also, because for all As is a star-homomorphism, , and we conclude that .

Proof of 1 (lifting to elements with same norm):
Let be an element in , and put Then is a self-adjoint element in , and Consult exercise 1.15 regarding the third equality sign. It follows from 2. that there exists a self-adjoint lift of with . The element in is a lift of .
By (1.4) as in the proof of 2., necessarily thus

Proof of 3 (positive elements do lift):
Let be any lift of , and set then and As in the proof of 2., put Then is normal, , and Hence is positive and .

Proof of 4
See exercise 9.4(2)
Proof of 5 (Projections do not lift):
Let , let , let and let , then is a projection in , and there is no projection in such that .
Proof of 6 (Unitaries don’t lift in general):
Exercise.

2.3 Semigroups of projections

Definition 2.3.1 (The semigroup of projections of A)
Put where is a C-star-Algebra and . We view the sets as being pair-wise disjoint.
Define the relation on as follows.
Suppose that is a projection in and is a projection in Then if and only if there exists an element with and . Here is the set of matrices with entries in . The adjoint is the entry-wise adjoint, and products are well-defined for these objects.

Define a binary operation on by ,
so that belongs to when and
The relation is an equivalence relation on . It combines the MvN equivalence with an identification of projections in different sized matrix algebras over . If if and only if and are MvN equivalent. The relation is named after the group it will help defining.

Proposition 2.3.2.
Let for some -algebra .

for , is the zero element in Identity element
If and , then coordinatewise transitivity
Symmetric
If are projections in such that , then is a projection and .
Asssociative

Definition 2.3.3. (The semigroup of projections)
With as in definition 2.3.1. set For each let in denote the equivalence class containing . Define addition on by for two projections in
It follows from Proposition 2.3.2. that this operation is well-defined and that is an abelian semigroup.
In the next chapter we will construct for each unital C*-algebra , an abelian group from the semigroup .